Predictability of extreme events in a nonlinear stochastic-dynamical model

Shock trace prediction by reduced models for a viscous stochastic Burgers equation

Viscous shocks are a particular type of extreme event in nonlinear multiscale systems, and their representation requires small scales. Model reduction can thus play an essential role in reducing the computational cost for the prediction of shocks. Yet, reduced models typically aim to approximate large-scale dominating dynamics, which do not resolve the small scales by design. To resolve this representation barrier, we introduce a new qualitative characterization of the space-time locations of shocks, named the "shock trace," via a space-time indicator function based on an empirical resolution-adaptive threshold. Unlike exact shocks, the shock traces can be captured within the representation capacity of the large scales, thus facilitating the forecast of the timing and locations of the shocks utilizing reduced models. Within the context of a viscous stochastic Burgers equation, we show that a data-driven reduced model, in the form of nonlinear autoregression (NAR) time series models, can accurately predict the random shock traces, with relatively low rates of false predictions. Furthermore, the NAR model, which includes nonlinear closure terms to approximate the feedback from the small scales, significantly outperforms the corresponding Galerkin truncated model in the scenario of either noiseless or noisy observations. The results illustrate the importance of the data-driven closure terms in the NAR model, which account for the effects of the unresolved dynamics brought by nonlinear interactions.

An efficient continuous data assimilation algorithm for the Sabra shell model of turbulence

Complex nonlinear turbulent dynamical systems are ubiquitous in many areas of research. Recovering unobserved state variables is an important topic for the data assimilation of turbulent systems. In this article, an efficient continuous-in-time data assimilation scheme is developed, which exploits closed analytic formulas for updating the unobserved state variables. Therefore, it is computationally efficient and accurate. The new data assimilation scheme is combined with a simple reduced order modeling technique that involves a cheap closure approximation and noise inflation. In such a way, many complicated turbulent dynamical systems can satisfy the requirements of the mathematical structures for the proposed efficient data assimilation scheme. The new data assimilation scheme is then applied to the Sabra shell model, which is a conceptual model for nonlinear turbulence. The goal is to recover the unobserved shell velocities across different spatial scales. It is shown that the new data assimilation scheme is skillful in capturing the nonlinear features of turbulence including the intermittency and extreme events in both the chaotic and the turbulent dynamical regimes. It is also shown that the new data assimilation scheme is more accurate and computationally cheaper than the standard ensemble Kalman filter and nudging data assimilation schemes for assimilating the Sabra shell model with partial observations.

Machine learning algorithms for predicting the amplitude of chaotic laser pulses

Forecasting the dynamics of chaotic systems from the analysis of their output signals is a challenging problem with applications in most fields of modern science. In this work, we use a laser model to compare the performance of several machine learning algorithms for forecasting the amplitude of upcoming emitted chaotic pulses. We simulate the dynamics of an optically injected semiconductor laser that presents a rich variety of dynamical regimes when changing the parameters. We focus on a particular dynamical regime that can show ultrahigh intensity pulses, reminiscent of rogue waves. We compare the goodness of the forecast for several popular methods in machine learning, namely, deep learning, support vector machine, nearest neighbors, and reservoir computing. Finally, we analyze how their performance for predicting the height of the next optical pulse depends on the amount of noise and the length of the time series used for training.

Effects of stochastic parametrization on extreme value statistics

Extreme geophysical events are of crucial relevance to our daily life: they threaten human lives and cause property damage. To assess the risk and reduce losses, we need to model and probabilistically predict these events. Parametrizations are computational tools used in the Earth system models, which are aimed at reproducing the impact of unresolved scales on resolved scales. The performance of parametrizations has usually been examined on typical events rather than on extreme events. In this paper, we consider a modified version of the two-level Lorenz'96 model and investigate how two parametrizations of the fast degrees of freedom perform in terms of the representation of extreme events. One parametrization is constructed following Wilks [Q. J. R. Meteorol. Soc. 131, 389-407 (2005)] and is constructed through an empirical fitting procedure; the other parametrization is constructed through the statistical mechanical approach proposed by Wouters and Lucarini [J. Stat. Mech. Theory Exp. 2012, P03003 (2012); J. Stat. Phys. 151, 850-860 (2013)]. The two strategies show different advantages and disadvantages. We discover that the agreement between parametrized models and true model is in general worse when looking at extremes rather than at the bulk of the statistics. The results suggest that stochastic parametrizations should be accurately and specifically tested against their performance on extreme events, as usual optimization procedures might neglect them.

Rare event simulation for steady-state probabilities via recurrency cycles

We develop a new algorithm for the estimation of rare event probabilities associated with the steady-state of a Markov stochastic process with continuous state space R^d and discrete time steps (i.e., a discrete-time R^d-valued Markov chain). The algorithm, which we coin Recurrent Multilevel Splitting (RMS), relies on the Markov chain's underlying recurrent structure, in combination with the Multilevel Splitting method. Extensive simulation experiments are performed, including experiments with a nonlinear stochastic model that has some characteristics of complex climate models. The numerical experiments show that RMS can boost the computational efficiency by several orders of magnitude compared to the Monte Carlo method.

Predictability of fat-tailed extremes

We conjecture for a linear stochastic differential equation that the predictability of threshold exceedances (I) improves with the event magnitude when the noise is a so-called correlated additive-multiplicative noise, no matter the nature of the stochastic innovations, and also improves when (II) the noise is purely additive, obeying a distribution that decays fast, i.e., not by a power law, and (III) deteriorates only when the additive noise distribution follows a power law. The predictability is measured by a summary index of the receiver operating characteristic curve. We provide support to our conjecture-to compliment reports in the existing literature on (II)-by a set of case studies. Calculations for the prediction skill are conducted in some cases by a direct numerical time-series-data-driven approach and in other cases by an analytical or semianalytical approach developed here.

Extremes in dynamic-stochastic systems

Extreme events capture the attention and imagination of the general public. Extreme events, especially meteorological and climatological extremes, cause significant economic damages and lead to a significant number of casualties each year. Thus, the prediction of extremes is of obvious importance. Here, I will survey the predictive skill and the predictability of extremes using dynamic-stochastic models. These dynamic-stochastic models combine deterministic nonlinear dynamics with a stochastic component, which consists potentially of both additive and multiplicative noise components. In these models, extremes are created by either the nonlinear dynamics, multiplicative noise, or additive heavy-tailed noises. These models naturally capture the observed clustering of extremes and can be used for the prediction of extremes.

Data-driven prediction and prevention of extreme events in a spatially extended excitable system

Extreme events occur in many spatially extended dynamical systems, often devastatingly affecting human life, which makes their reliable prediction and efficient prevention highly desirable. We study the prediction and prevention of extreme events in a spatially extended system, a system of coupled FitzHugh-Nagumo units, in which extreme events occur in a spatially and temporally irregular way. Mimicking typical constraints faced in field studies, we assume not to know the governing equations of motion and to be able to observe only a subset of all phase-space variables for a limited period of time. Based on reconstructing the local dynamics from data and despite being challenged by the rareness of events, we are able to predict extreme events remarkably well. With small, rare, and spatiotemporally localized perturbations which are guided by our predictions, we are able to completely suppress extreme events in this system.

Finite-size effects on return interval distributions for weakest-link-scaling systems

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the κ-Weibull distribution. The upper tail of the κ-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the κ-Weibull distribution decreases linearly after a waiting time τ(c) ∝ n(1/m), where m is the Weibull modulus and n is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the κ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the κ-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

Persistent regimes and extreme events of the North Atlantic atmospheric circulation

Society is increasingly impacted by natural hazards which cause significant damage in economic and human terms. Many of these natural hazards are weather and climate related. Here, we show that North Atlantic atmospheric circulation regimes affect the propensity of extreme wind speeds in Europe. We also show evidence that extreme wind speeds are long-range dependent, follow a generalized Pareto distribution and are serially clustered. Serial clustering means that storms come in bunches and, hence, do not occur independently. We discuss the use of waiting time distributions for extreme event recurrence estimation in serially dependent time series.